Properties

Label 2-177-59.58-c2-0-8
Degree $2$
Conductor $177$
Sign $0.904 + 0.425i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.537i·2-s − 1.73·3-s + 3.71·4-s − 0.803·5-s + 0.931i·6-s + 5.11·7-s − 4.14i·8-s + 2.99·9-s + 0.431i·10-s + 17.7i·11-s − 6.42·12-s − 24.6i·13-s − 2.74i·14-s + 1.39·15-s + 12.6·16-s + 18.2·17-s + ⋯
L(s)  = 1  − 0.268i·2-s − 0.577·3-s + 0.927·4-s − 0.160·5-s + 0.155i·6-s + 0.730·7-s − 0.518i·8-s + 0.333·9-s + 0.0431i·10-s + 1.61i·11-s − 0.535·12-s − 1.89i·13-s − 0.196i·14-s + 0.0927·15-s + 0.788·16-s + 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.904 + 0.425i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ 0.904 + 0.425i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.61165 - 0.360069i\)
\(L(\frac12)\) \(\approx\) \(1.61165 - 0.360069i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
59 \( 1 + (53.3 + 25.1i)T \)
good2 \( 1 + 0.537iT - 4T^{2} \)
5 \( 1 + 0.803T + 25T^{2} \)
7 \( 1 - 5.11T + 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + 24.6iT - 169T^{2} \)
17 \( 1 - 18.2T + 289T^{2} \)
19 \( 1 - 28.5T + 361T^{2} \)
23 \( 1 + 11.8iT - 529T^{2} \)
29 \( 1 - 9.01T + 841T^{2} \)
31 \( 1 + 4.94iT - 961T^{2} \)
37 \( 1 - 39.7iT - 1.36e3T^{2} \)
41 \( 1 + 38.0T + 1.68e3T^{2} \)
43 \( 1 + 19.2iT - 1.84e3T^{2} \)
47 \( 1 - 65.6iT - 2.20e3T^{2} \)
53 \( 1 + 40.1T + 2.80e3T^{2} \)
61 \( 1 - 110. iT - 3.72e3T^{2} \)
67 \( 1 + 30.7iT - 4.48e3T^{2} \)
71 \( 1 + 95.0T + 5.04e3T^{2} \)
73 \( 1 - 71.2iT - 5.32e3T^{2} \)
79 \( 1 + 13.3T + 6.24e3T^{2} \)
83 \( 1 + 142. iT - 6.88e3T^{2} \)
89 \( 1 + 128. iT - 7.92e3T^{2} \)
97 \( 1 + 97.1iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.13356196427521252157611877632, −11.59653208706668374889451381934, −10.34384097897141084342002711052, −9.937007287712751980073594287226, −7.84383269126845759568953662827, −7.40466907510709219626830340166, −5.90899474051703746998813374993, −4.86763672207425459506311760115, −3.09269846904502707001960919090, −1.38030843357097825901642573443, 1.52834207301522552586724285155, 3.49225204699854748498671316408, 5.23283628239917742397739038836, 6.15075266629380534092813964451, 7.27172727887612883245324189386, 8.201047591896286582999821476067, 9.568261286597801914033562858337, 10.93485037320425087682255257650, 11.60422634462394494579203282345, 11.97859003141396509261485501326

Graph of the $Z$-function along the critical line